I don't really know the answer, but I suppose I would first start by trying to disprove the conjecture. After all, it has only been verified for graphs up to order 5. The obvious counterexamples I would check are large cliques and large anti-cliques.
So, do there exist arbitrary long sequences of consecutive demi-primes that are pairwise co-prime? What about arbitrary long sequences of consecutive demi-primes such that each pair has a common factor?
The number theorists can feel free to chime in here anytime.
If those don't work, then some other candidates for counterexamples would be large matchings or large cliques together with an isolated vertex.
Edit: I just read that it is strongly believed that there are arbitrarily long sequences of consecutive primes such that each prime is congruent to 3 (mod 4). If true, this would give a representation of arbitrarily large anti-cliques, since the corresponding sequence of demi-primes would all be even. Does anyone know if this has been proven?